LMFDB - Embedded newform 8037.2.a.w.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0899584 q^{2} -1.99191 q^{4} -1.20146 q^{5} +3.43586 q^{7} -0.359105 q^{8} +O(q^{10})$	\(q+0.0899584 q^{2} -1.99191 q^{4} -1.20146 q^{5} +3.43586 q^{7} -0.359105 q^{8} -0.108081 q^{10} +0.0198204 q^{11} +4.14315 q^{13} +0.309085 q^{14} +3.95151 q^{16} +4.43516 q^{17} -1.00000 q^{19} +2.39320 q^{20} +0.00178301 q^{22} -1.27631 q^{23} -3.55649 q^{25} +0.372711 q^{26} -6.84392 q^{28} +7.52638 q^{29} -3.98857 q^{31} +1.07368 q^{32} +0.398980 q^{34} -4.12806 q^{35} +1.29159 q^{37} -0.0899584 q^{38} +0.431451 q^{40} +9.40722 q^{41} +6.68193 q^{43} -0.0394803 q^{44} -0.114815 q^{46} +1.00000 q^{47} +4.80516 q^{49} -0.319936 q^{50} -8.25277 q^{52} +13.7254 q^{53} -0.0238134 q^{55} -1.23384 q^{56} +0.677060 q^{58} -8.51100 q^{59} -1.65744 q^{61} -0.358806 q^{62} -7.80643 q^{64} -4.97783 q^{65} -12.5242 q^{67} -8.83444 q^{68} -0.371353 q^{70} +9.46292 q^{71} -10.6135 q^{73} +0.116190 q^{74} +1.99191 q^{76} +0.0681001 q^{77} +7.20351 q^{79} -4.74758 q^{80} +0.846258 q^{82} +13.0596 q^{83} -5.32867 q^{85} +0.601095 q^{86} -0.00711760 q^{88} +5.10160 q^{89} +14.2353 q^{91} +2.54229 q^{92} +0.0899584 q^{94} +1.20146 q^{95} -1.40864 q^{97} +0.432265 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0899584	0.0636102	0.0318051	−	0.999494i	$-0.489874\pi$
$2$	0.0899584	0.0636102	0.0318051	+	0.999494i	$0.489874\pi$
$3$	0	0
$3$	0	0
$4$	−1.99191	−0.995954
$5$	−1.20146	−0.537309	−0.268655	−	0.963237i	$-0.586579\pi$
$5$	−1.20146	−0.537309	−0.268655	+	0.963237i	$0.586579\pi$
$6$	0	0
$7$	3.43586	1.29863	0.649317	−	0.760518i	$-0.275055\pi$
$7$	3.43586	1.29863	0.649317	+	0.760518i	$0.275055\pi$
$8$	−0.359105	−0.126963
$9$	0	0
$10$	−0.108081	−0.0341783
$11$	0.0198204	0.00597606	0.00298803	−	0.999996i	$-0.499049\pi$
$11$	0.0198204	0.00597606	0.00298803	+	0.999996i	$0.499049\pi$
$12$	0	0
$13$	4.14315	1.14910	0.574551	−	0.818469i	$-0.305177\pi$
$13$	4.14315	1.14910	0.574551	+	0.818469i	$0.305177\pi$
$14$	0.309085	0.0826064
$15$	0	0
$16$	3.95151	0.987878
$17$	4.43516	1.07569	0.537843	−	0.843045i	$-0.319239\pi$
$17$	4.43516	1.07569	0.537843	+	0.843045i	$0.319239\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.39320	0.535135
$21$	0	0
$22$	0.00178301	0.000380138 0
$23$	−1.27631	−0.266129	−0.133065	−	0.991107i	$-0.542482\pi$
$23$	−1.27631	−0.266129	−0.133065	+	0.991107i	$0.542482\pi$
$24$	0	0
$25$	−3.55649	−0.711299
$26$	0.372711	0.0730946
$27$	0	0
$28$	−6.84392	−1.29338
$29$	7.52638	1.39761	0.698806	−	0.715311i	$-0.253715\pi$
$29$	7.52638	1.39761	0.698806	+	0.715311i	$0.253715\pi$
$30$	0	0
$31$	−3.98857	−0.716369	−0.358184	−	0.933651i	$-0.616604\pi$
$31$	−3.98857	−0.716369	−0.358184	+	0.933651i	$0.616604\pi$
$32$	1.07368	0.189802
$33$	0	0
$34$	0.398980	0.0684245
$35$	−4.12806	−0.697769
$36$	0	0
$37$	1.29159	0.212337	0.106168	−	0.994348i	$-0.466142\pi$
$37$	1.29159	0.212337	0.106168	+	0.994348i	$0.466142\pi$
$38$	−0.0899584	−0.0145932
$39$	0	0
$40$	0.431451	0.0682184
$41$	9.40722	1.46916	0.734580	−	0.678522i	$-0.237379\pi$
$41$	9.40722	1.46916	0.734580	+	0.678522i	$0.237379\pi$
$42$	0	0
$43$	6.68193	1.01898	0.509492	−	0.860475i	$-0.329833\pi$
$43$	6.68193	1.01898	0.509492	+	0.860475i	$0.329833\pi$
$44$	−0.0394803	−0.00595188
$45$	0	0
$46$	−0.114815	−0.0169285
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	4.80516	0.686452
$50$	−0.319936	−0.0452458
$51$	0	0
$52$	−8.25277	−1.14445
$53$	13.7254	1.88533	0.942664	−	0.333744i	$-0.108312\pi$
$53$	13.7254	1.88533	0.942664	+	0.333744i	$0.108312\pi$
$54$	0	0
$55$	−0.0238134	−0.00321099
$56$	−1.23384	−0.164878
$57$	0	0
$58$	0.677060	0.0889024
$59$	−8.51100	−1.10804	−0.554019	−	0.832504i	$-0.686907\pi$
$59$	−8.51100	−1.10804	−0.554019	+	0.832504i	$0.686907\pi$
$60$	0	0
$61$	−1.65744	−0.212213	−0.106106	−	0.994355i	$-0.533838\pi$
$61$	−1.65744	−0.212213	−0.106106	+	0.994355i	$0.533838\pi$
$62$	−0.358806	−0.0455683
$63$	0	0
$64$	−7.80643	−0.975804
$65$	−4.97783	−0.617424
$66$	0	0
$67$	−12.5242	−1.53007	−0.765036	−	0.643988i	$-0.777279\pi$
$67$	−12.5242	−1.53007	−0.765036	+	0.643988i	$0.777279\pi$
$68$	−8.83444	−1.07133
$69$	0	0
$70$	−0.371353	−0.0443852
$71$	9.46292	1.12304	0.561521	−	0.827462i	$-0.310216\pi$
$71$	9.46292	1.12304	0.561521	+	0.827462i	$0.310216\pi$
$72$	0	0
$73$	−10.6135	−1.24222	−0.621110	−	0.783724i	$-0.713318\pi$
$73$	−10.6135	−1.24222	−0.621110	+	0.783724i	$0.713318\pi$
$74$	0.116190	0.0135068
$75$	0	0
$76$	1.99191	0.228487
$77$	0.0681001	0.00776072
$78$	0	0
$79$	7.20351	0.810459	0.405229	−	0.914215i	$-0.367192\pi$
$79$	7.20351	0.810459	0.405229	+	0.914215i	$0.367192\pi$
$80$	−4.74758	−0.530796
$81$	0	0
$82$	0.846258	0.0934536
$83$	13.0596	1.43348	0.716740	−	0.697341i	$-0.245634\pi$
$83$	13.0596	1.43348	0.716740	+	0.697341i	$0.245634\pi$
$84$	0	0
$85$	−5.32867	−0.577976
$86$	0.601095	0.0648178
$87$	0	0
$88$	−0.00711760	−0.000758738 0
$89$	5.10160	0.540769	0.270384	−	0.962752i	$-0.412849\pi$
$89$	5.10160	0.540769	0.270384	+	0.962752i	$0.412849\pi$
$90$	0	0
$91$	14.2353	1.49226
$92$	2.54229	0.265052
$93$	0	0
$94$	0.0899584	0.00927850
$95$	1.20146	0.123267
$96$	0	0
$97$	−1.40864	−0.143026	−0.0715130	−	0.997440i	$-0.522783\pi$
$97$	−1.40864	−0.143026	−0.0715130	+	0.997440i	$0.522783\pi$
$98$	0.432265	0.0436653
$99$	0	0
$100$	7.08420	0.708420
$101$	−9.65251	−0.960460	−0.480230	−	0.877143i	$-0.659447\pi$
$101$	−9.65251	−0.960460	−0.480230	+	0.877143i	$0.659447\pi$
$102$	0	0
$103$	−6.71296	−0.661448	−0.330724	−	0.943728i	$-0.607293\pi$
$103$	−6.71296	−0.661448	−0.330724	+	0.943728i	$0.607293\pi$
$104$	−1.48783	−0.145893
$105$	0	0
$106$	1.23471	0.119926
$107$	−0.305153	−0.0295002	−0.0147501	−	0.999891i	$-0.504695\pi$
$107$	−0.305153	−0.0295002	−0.0147501	+	0.999891i	$0.504695\pi$
$108$	0	0
$109$	−19.2788	−1.84657	−0.923287	−	0.384110i	$-0.874508\pi$
$109$	−19.2788	−1.84657	−0.923287	+	0.384110i	$0.874508\pi$
$110$	−0.00214221	−0.000204252 0
$111$	0	0
$112$	13.5769	1.28289
$113$	−5.05217	−0.475269	−0.237634	−	0.971355i	$-0.576372\pi$
$113$	−5.05217	−0.475269	−0.237634	+	0.971355i	$0.576372\pi$
$114$	0	0
$115$	1.53344	0.142994
$116$	−14.9918	−1.39196
$117$	0	0
$118$	−0.765636	−0.0704825
$119$	15.2386	1.39692
$120$	0	0
$121$	−10.9996	−0.999964
$122$	−0.149100	−0.0134989
$123$	0	0
$124$	7.94487	0.713470
$125$	10.2803	0.919497
$126$	0	0
$127$	7.20979	0.639765	0.319883	−	0.947457i	$-0.396357\pi$
$127$	7.20979	0.639765	0.319883	+	0.947457i	$0.396357\pi$
$128$	−2.84962	−0.251873
$129$	0	0
$130$	−0.447797	−0.0392744
$131$	−13.1889	−1.15232	−0.576162	−	0.817336i	$-0.695450\pi$
$131$	−13.1889	−1.15232	−0.576162	+	0.817336i	$0.695450\pi$
$132$	0	0
$133$	−3.43586	−0.297927
$134$	−1.12665	−0.0973281
$135$	0	0
$136$	−1.59269	−0.136572
$137$	0.0701165	0.00599046	0.00299523	−	0.999996i	$-0.499047\pi$
$137$	0.0701165	0.00599046	0.00299523	+	0.999996i	$0.499047\pi$
$138$	0	0
$139$	5.72735	0.485788	0.242894	−	0.970053i	$-0.421903\pi$
$139$	5.72735	0.485788	0.242894	+	0.970053i	$0.421903\pi$
$140$	8.22270	0.694945
$141$	0	0
$142$	0.851269	0.0714369
$143$	0.0821187	0.00686711
$144$	0	0
$145$	−9.04264	−0.750951
$146$	−0.954775	−0.0790178
$147$	0	0
$148$	−2.57274	−0.211478
$149$	−13.7901	−1.12973	−0.564864	−	0.825184i	$-0.691071\pi$
$149$	−13.7901	−1.12973	−0.564864	+	0.825184i	$0.691071\pi$
$150$	0	0
$151$	−7.86308	−0.639888	−0.319944	−	0.947436i	$-0.603664\pi$
$151$	−7.86308	−0.639888	−0.319944	+	0.947436i	$0.603664\pi$
$152$	0.359105	0.0291273
$153$	0	0
$154$	0.00612617	0.000493661 0
$155$	4.79211	0.384912
$156$	0	0
$157$	18.5221	1.47822	0.739112	−	0.673583i	$-0.235245\pi$
$157$	18.5221	1.47822	0.739112	+	0.673583i	$0.235245\pi$
$158$	0.648016	0.0515534
$159$	0	0
$160$	−1.28999	−0.101982
$161$	−4.38523	−0.345605
$162$	0	0
$163$	17.6670	1.38378	0.691892	−	0.722001i	$-0.256777\pi$
$163$	17.6670	1.38378	0.691892	+	0.722001i	$0.256777\pi$
$164$	−18.7383	−1.46322
$165$	0	0
$166$	1.17482	0.0911839
$167$	−10.3420	−0.800290	−0.400145	−	0.916452i	$-0.631040\pi$
$167$	−10.3420	−0.800290	−0.400145	+	0.916452i	$0.631040\pi$
$168$	0	0
$169$	4.16568	0.320437
$170$	−0.479359	−0.0367651
$171$	0	0
$172$	−13.3098	−1.01486
$173$	−22.5253	−1.71256	−0.856282	−	0.516509i	$-0.827231\pi$
$173$	−22.5253	−1.71256	−0.856282	+	0.516509i	$0.827231\pi$
$174$	0	0
$175$	−12.2196	−0.923717
$176$	0.0783203	0.00590362
$177$	0	0
$178$	0.458932	0.0343984
$179$	−6.32827	−0.472997	−0.236499	−	0.971632i	$-0.576000\pi$
$179$	−6.32827	−0.472997	−0.236499	+	0.971632i	$0.576000\pi$
$180$	0	0
$181$	−0.506637	−0.0376581	−0.0188290	−	0.999823i	$-0.505994\pi$
$181$	−0.506637	−0.0376581	−0.0188290	+	0.999823i	$0.505994\pi$
$182$	1.28058	0.0949232
$183$	0	0
$184$	0.458330	0.0337886
$185$	−1.55180	−0.114091
$186$	0	0
$187$	0.0879065	0.00642836
$188$	−1.99191	−0.145275
$189$	0	0
$190$	0.108081	0.00784105
$191$	20.1528	1.45820	0.729101	−	0.684406i	$-0.239938\pi$
$191$	20.1528	1.45820	0.729101	+	0.684406i	$0.239938\pi$
$192$	0	0
$193$	−14.7927	−1.06480	−0.532400	−	0.846493i	$-0.678710\pi$
$193$	−14.7927	−1.06480	−0.532400	+	0.846493i	$0.678710\pi$
$194$	−0.126719	−0.00909790
$195$	0	0
$196$	−9.57144	−0.683675
$197$	−4.86138	−0.346359	−0.173179	−	0.984890i	$-0.555404\pi$
$197$	−4.86138	−0.346359	−0.173179	+	0.984890i	$0.555404\pi$
$198$	0	0
$199$	28.0571	1.98891	0.994457	−	0.105145i	$-0.0335306\pi$
$199$	28.0571	1.98891	0.994457	+	0.105145i	$0.0335306\pi$
$200$	1.27716	0.0903086
$201$	0	0
$202$	−0.868324	−0.0610950
$203$	25.8596	1.81499
$204$	0	0
$205$	−11.3024	−0.789394
$206$	−0.603887	−0.0420748
$207$	0	0
$208$	16.3717	1.13517
$209$	−0.0198204	−0.00137100
$210$	0	0
$211$	5.21992	0.359354	0.179677	−	0.983726i	$-0.442495\pi$
$211$	5.21992	0.359354	0.179677	+	0.983726i	$0.442495\pi$
$212$	−27.3397	−1.87770
$213$	0	0
$214$	−0.0274510	−0.00187651
$215$	−8.02807	−0.547510
$216$	0	0
$217$	−13.7042	−0.930302
$218$	−1.73429	−0.117461
$219$	0	0
$220$	0.0474340	0.00319800
$221$	18.3755	1.23607
$222$	0	0
$223$	26.5203	1.77593	0.887965	−	0.459912i	$-0.152119\pi$
$223$	26.5203	1.77593	0.887965	+	0.459912i	$0.152119\pi$
$224$	3.68903	0.246483
$225$	0	0
$226$	−0.454485	−0.0302319
$227$	1.58451	0.105168	0.0525838	−	0.998617i	$-0.483254\pi$
$227$	1.58451	0.105168	0.0525838	+	0.998617i	$0.483254\pi$
$228$	0	0
$229$	4.47336	0.295608	0.147804	−	0.989017i	$-0.452780\pi$
$229$	4.47336	0.295608	0.147804	+	0.989017i	$0.452780\pi$
$230$	0.137945	0.00909586
$231$	0	0
$232$	−2.70276	−0.177445
$233$	6.28755	0.411911	0.205955	−	0.978561i	$-0.433970\pi$
$233$	6.28755	0.411911	0.205955	+	0.978561i	$0.433970\pi$
$234$	0	0
$235$	−1.20146	−0.0783746
$236$	16.9531	1.10355
$237$	0	0
$238$	1.37084	0.0888585
$239$	7.92540	0.512651	0.256326	−	0.966590i	$-0.417488\pi$
$239$	7.92540	0.512651	0.256326	+	0.966590i	$0.417488\pi$
$240$	0	0
$241$	−16.6702	−1.07382	−0.536910	−	0.843639i	$-0.680409\pi$
$241$	−16.6702	−1.07382	−0.536910	+	0.843639i	$0.680409\pi$
$242$	−0.989507	−0.0636079
$243$	0	0
$244$	3.30146	0.211354
$245$	−5.77322	−0.368837
$246$	0	0
$247$	−4.14315	−0.263622
$248$	1.43232	0.0909523
$249$	0	0
$250$	0.924798	0.0584893
$251$	24.0152	1.51583	0.757913	−	0.652355i	$-0.226219\pi$
$251$	24.0152	1.51583	0.757913	+	0.652355i	$0.226219\pi$
$252$	0	0
$253$	−0.0252969	−0.00159040
$254$	0.648581	0.0406956
$255$	0	0
$256$	15.3565	0.959783
$257$	21.9355	1.36830	0.684150	−	0.729342i	$-0.260174\pi$
$257$	21.9355	1.36830	0.684150	+	0.729342i	$0.260174\pi$
$258$	0	0
$259$	4.43774	0.275748
$260$	9.91537	0.614925
$261$	0	0
$262$	−1.18646	−0.0732995
$263$	29.4867	1.81822	0.909112	−	0.416551i	$-0.136761\pi$
$263$	29.4867	1.81822	0.909112	+	0.416551i	$0.136761\pi$
$264$	0	0
$265$	−16.4905	−1.01300
$266$	−0.309085	−0.0189512
$267$	0	0
$268$	24.9470	1.52388
$269$	13.6946	0.834976	0.417488	−	0.908682i	$-0.362911\pi$
$269$	13.6946	0.834976	0.417488	+	0.908682i	$0.362911\pi$
$270$	0	0
$271$	7.93923	0.482274	0.241137	−	0.970491i	$-0.422480\pi$
$271$	7.93923	0.482274	0.241137	+	0.970491i	$0.422480\pi$
$272$	17.5256	1.06265
$273$	0	0
$274$	0.00630756	0.000381054 0
$275$	−0.0704910	−0.00425076
$276$	0	0
$277$	13.9356	0.837309	0.418654	−	0.908146i	$-0.362502\pi$
$277$	13.9356	0.837309	0.418654	+	0.908146i	$0.362502\pi$
$278$	0.515223	0.0309010
$279$	0	0
$280$	1.48241	0.0885908
$281$	14.0784	0.839844	0.419922	−	0.907560i	$-0.362057\pi$
$281$	14.0784	0.839844	0.419922	+	0.907560i	$0.362057\pi$
$282$	0	0
$283$	−10.4128	−0.618976	−0.309488	−	0.950903i	$-0.600158\pi$
$283$	−10.4128	−0.618976	−0.309488	+	0.950903i	$0.600158\pi$
$284$	−18.8493	−1.11850
$285$	0	0
$286$	0.00738726	0.000436818 0
$287$	32.3219	1.90790
$288$	0	0
$289$	2.67068	0.157099
$290$	−0.813461	−0.0477681
$291$	0	0
$292$	21.1412	1.23719
$293$	30.8393	1.80165	0.900825	−	0.434181i	$-0.142962\pi$
$293$	30.8393	1.80165	0.900825	+	0.434181i	$0.142962\pi$
$294$	0	0
$295$	10.2256	0.595359
$296$	−0.463818	−0.0269589
$297$	0	0
$298$	−1.24053	−0.0718622
$299$	−5.28795	−0.305810
$300$	0	0
$301$	22.9582	1.32329
$302$	−0.707350	−0.0407034
$303$	0	0
$304$	−3.95151	−0.226635
$305$	1.99134	0.114024
$306$	0	0
$307$	4.97753	0.284083	0.142041	−	0.989861i	$-0.454633\pi$
$307$	4.97753	0.284083	0.142041	+	0.989861i	$0.454633\pi$
$308$	−0.135649	−0.00772932
$309$	0	0
$310$	0.431091	0.0244843
$311$	1.95132	0.110649	0.0553246	−	0.998468i	$-0.482381\pi$
$311$	1.95132	0.110649	0.0553246	+	0.998468i	$0.482381\pi$
$312$	0	0
$313$	13.0327	0.736651	0.368325	−	0.929697i	$-0.379931\pi$
$313$	13.0327	0.736651	0.368325	+	0.929697i	$0.379931\pi$
$314$	1.66622	0.0940301
$315$	0	0
$316$	−14.3487	−0.807179
$317$	15.8116	0.888066	0.444033	−	0.896010i	$-0.353547\pi$
$317$	15.8116	0.888066	0.444033	+	0.896010i	$0.353547\pi$
$318$	0	0
$319$	0.149175	0.00835222
$320$	9.37912	0.524309
$321$	0	0
$322$	−0.394488	−0.0219840
$323$	−4.43516	−0.246779
$324$	0	0
$325$	−14.7351	−0.817355
$326$	1.58929	0.0880227
$327$	0	0
$328$	−3.37818	−0.186529
$329$	3.43586	0.189425
$330$	0	0
$331$	6.98290	0.383815	0.191908	−	0.981413i	$-0.438533\pi$
$331$	6.98290	0.383815	0.191908	+	0.981413i	$0.438533\pi$
$332$	−26.0136	−1.42768
$333$	0	0
$334$	−0.930351	−0.0509066
$335$	15.0473	0.822122
$336$	0	0
$337$	−19.8618	−1.08194	−0.540970	−	0.841042i	$-0.681943\pi$
$337$	−19.8618	−1.08194	−0.540970	+	0.841042i	$0.681943\pi$
$338$	0.374738	0.0203830
$339$	0	0
$340$	10.6142	0.575637
$341$	−0.0790549	−0.00428107
$342$	0	0
$343$	−7.54116	−0.407184
$344$	−2.39952	−0.129373
$345$	0	0
$346$	−2.02634	−0.108936
$347$	−29.6356	−1.59093	−0.795463	−	0.606003i	$-0.792772\pi$
$347$	−29.6356	−1.59093	−0.795463	+	0.606003i	$0.792772\pi$
$348$	0	0
$349$	−15.4903	−0.829179	−0.414589	−	0.910009i	$-0.636075\pi$
$349$	−15.4903	−0.829179	−0.414589	+	0.910009i	$0.636075\pi$
$350$	−1.09926	−0.0587578
$351$	0	0
$352$	0.0212808	0.00113427
$353$	8.08080	0.430098	0.215049	−	0.976603i	$-0.431009\pi$
$353$	8.08080	0.430098	0.215049	+	0.976603i	$0.431009\pi$
$354$	0	0
$355$	−11.3693	−0.603421
$356$	−10.1619	−0.538580
$357$	0	0
$358$	−0.569281	−0.0300874
$359$	−5.22082	−0.275544	−0.137772	−	0.990464i	$-0.543994\pi$
$359$	−5.22082	−0.275544	−0.137772	+	0.990464i	$0.543994\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.0455763	−0.00239544
$363$	0	0
$364$	−28.3554	−1.48623
$365$	12.7517	0.667456
$366$	0	0
$367$	−21.9858	−1.14765	−0.573825	−	0.818978i	$-0.694541\pi$
$367$	−21.9858	−1.14765	−0.573825	+	0.818978i	$0.694541\pi$
$368$	−5.04336	−0.262903
$369$	0	0
$370$	−0.139597	−0.00725732
$371$	47.1586	2.44835
$372$	0	0
$373$	7.81504	0.404647	0.202324	−	0.979319i	$-0.435151\pi$
$373$	7.81504	0.404647	0.202324	+	0.979319i	$0.435151\pi$
$374$	0.00790793	0.000408909 0
$375$	0	0
$376$	−0.359105	−0.0185194
$377$	31.1829	1.60600
$378$	0	0
$379$	−0.242508	−0.0124568	−0.00622840	−	0.999981i	$-0.501983\pi$
$379$	−0.242508	−0.0124568	−0.00622840	+	0.999981i	$0.501983\pi$
$380$	−2.39320	−0.122768
$381$	0	0
$382$	1.81291	0.0927565
$383$	−29.6637	−1.51575	−0.757873	−	0.652402i	$-0.773761\pi$
$383$	−29.6637	−1.51575	−0.757873	+	0.652402i	$0.773761\pi$
$384$	0	0
$385$	−0.0818195	−0.00416991
$386$	−1.33072	−0.0677321
$387$	0	0
$388$	2.80588	0.142447
$389$	−7.61500	−0.386096	−0.193048	−	0.981189i	$-0.561837\pi$
$389$	−7.61500	−0.386096	−0.193048	+	0.981189i	$0.561837\pi$
$390$	0	0
$391$	−5.66065	−0.286271
$392$	−1.72556	−0.0871540
$393$	0	0
$394$	−0.437321	−0.0220319
$395$	−8.65474	−0.435467
$396$	0	0
$397$	−0.578622	−0.0290402	−0.0145201	−	0.999895i	$-0.504622\pi$
$397$	−0.578622	−0.0290402	−0.0145201	+	0.999895i	$0.504622\pi$
$398$	2.52397	0.126515
$399$	0	0
$400$	−14.0535	−0.702676
$401$	−3.15371	−0.157489	−0.0787444	−	0.996895i	$-0.525091\pi$
$401$	−3.15371	−0.157489	−0.0787444	+	0.996895i	$0.525091\pi$
$402$	0	0
$403$	−16.5253	−0.823181
$404$	19.2269	0.956574
$405$	0	0
$406$	2.32629	0.115452
$407$	0.0255999	0.00126894
$408$	0	0
$409$	17.5849	0.869519	0.434759	−	0.900547i	$-0.356833\pi$
$409$	17.5849	0.869519	0.434759	+	0.900547i	$0.356833\pi$
$410$	−1.01675	−0.0502135
$411$	0	0
$412$	13.3716	0.658771
$413$	−29.2426	−1.43894
$414$	0	0
$415$	−15.6906	−0.770222
$416$	4.44843	0.218102
$417$	0	0
$418$	−0.00178301	−8.72097e−5 0
$419$	31.7745	1.55228	0.776142	−	0.630558i	$-0.217174\pi$
$419$	31.7745	1.55228	0.776142	+	0.630558i	$0.217174\pi$
$420$	0	0
$421$	31.5437	1.53734	0.768672	−	0.639643i	$-0.220918\pi$
$421$	31.5437	1.53734	0.768672	+	0.639643i	$0.220918\pi$
$422$	0.469575	0.0228586
$423$	0	0
$424$	−4.92886	−0.239367
$425$	−15.7736	−0.765133
$426$	0	0
$427$	−5.69473	−0.275587
$428$	0.607836	0.0293808
$429$	0	0
$430$	−0.722192	−0.0348272
$431$	12.4313	0.598795	0.299397	−	0.954129i	$-0.403214\pi$
$431$	12.4313	0.598795	0.299397	+	0.954129i	$0.403214\pi$
$432$	0	0
$433$	18.0072	0.865369	0.432685	−	0.901545i	$-0.357566\pi$
$433$	18.0072	0.865369	0.432685	+	0.901545i	$0.357566\pi$
$434$	−1.23281	−0.0591766
$435$	0	0
$436$	38.4016	1.83910
$437$	1.27631	0.0610542
$438$	0	0
$439$	2.34017	0.111690	0.0558452	−	0.998439i	$-0.482215\pi$
$439$	2.34017	0.111690	0.0558452	+	0.998439i	$0.482215\pi$
$440$	0.00855151	0.000407677 0
$441$	0	0
$442$	1.65303	0.0786268
$443$	9.34385	0.443939	0.221970	−	0.975054i	$-0.428751\pi$
$443$	9.34385	0.443939	0.221970	+	0.975054i	$0.428751\pi$
$444$	0	0
$445$	−6.12937	−0.290560
$446$	2.38572	0.112967
$447$	0	0
$448$	−26.8219	−1.26721
$449$	−13.4137	−0.633031	−0.316515	−	0.948587i	$-0.602513\pi$
$449$	−13.4137	−0.633031	−0.316515	+	0.948587i	$0.602513\pi$
$450$	0	0
$451$	0.186454	0.00877980
$452$	10.0635	0.473345
$453$	0	0
$454$	0.142540	0.00668973
$455$	−17.1031	−0.801808
$456$	0	0
$457$	19.0201	0.889724	0.444862	−	0.895599i	$-0.353253\pi$
$457$	19.0201	0.889724	0.444862	+	0.895599i	$0.353253\pi$
$458$	0.402416	0.0188037
$459$	0	0
$460$	−3.05446	−0.142415
$461$	−2.57229	−0.119803	−0.0599017	−	0.998204i	$-0.519079\pi$
$461$	−2.57229	−0.119803	−0.0599017	+	0.998204i	$0.519079\pi$
$462$	0	0
$463$	31.8696	1.48111	0.740553	−	0.671998i	$-0.234563\pi$
$463$	31.8696	1.48111	0.740553	+	0.671998i	$0.234563\pi$
$464$	29.7406	1.38067
$465$	0	0
$466$	0.565617	0.0262017
$467$	22.3272	1.03318	0.516589	−	0.856233i	$-0.327201\pi$
$467$	22.3272	1.03318	0.516589	+	0.856233i	$0.327201\pi$
$468$	0	0
$469$	−43.0314	−1.98700
$470$	−0.108081	−0.00498542
$471$	0	0
$472$	3.05635	0.140680
$473$	0.132438	0.00608952
$474$	0	0
$475$	3.55649	0.163183
$476$	−30.3539	−1.39127
$477$	0	0
$478$	0.712956	0.0326098
$479$	29.1063	1.32990	0.664951	−	0.746887i	$-0.268452\pi$
$479$	29.1063	1.32990	0.664951	+	0.746887i	$0.268452\pi$
$480$	0	0
$481$	5.35127	0.243997
$482$	−1.49962	−0.0683059
$483$	0	0
$484$	21.9102	0.995918
$485$	1.69243	0.0768492
$486$	0	0
$487$	−1.01534	−0.0460093	−0.0230047	−	0.999735i	$-0.507323\pi$
$487$	−1.01534	−0.0460093	−0.0230047	+	0.999735i	$0.507323\pi$
$488$	0.595194	0.0269432
$489$	0	0
$490$	−0.519349	−0.0234618
$491$	14.3345	0.646908	0.323454	−	0.946244i	$-0.395156\pi$
$491$	14.3345	0.646908	0.323454	+	0.946244i	$0.395156\pi$
$492$	0	0
$493$	33.3807	1.50339
$494$	−0.372711	−0.0167691
$495$	0	0
$496$	−15.7609	−0.707685
$497$	32.5133	1.45842
$498$	0	0
$499$	−14.4725	−0.647880	−0.323940	−	0.946078i	$-0.605008\pi$
$499$	−14.4725	−0.647880	−0.323940	+	0.946078i	$0.605008\pi$
$500$	−20.4774	−0.915776
$501$	0	0
$502$	2.16037	0.0964220
$503$	0.646476	0.0288249	0.0144125	−	0.999896i	$-0.495412\pi$
$503$	0.646476	0.0288249	0.0144125	+	0.999896i	$0.495412\pi$
$504$	0	0
$505$	11.5971	0.516064
$506$	−0.00227567	−0.000101166 0
$507$	0	0
$508$	−14.3612	−0.637177
$509$	10.2840	0.455828	0.227914	−	0.973681i	$-0.426809\pi$
$509$	10.2840	0.455828	0.227914	+	0.973681i	$0.426809\pi$
$510$	0	0
$511$	−36.4666	−1.61319
$512$	7.08068	0.312925
$513$	0	0
$514$	1.97328	0.0870377
$515$	8.06536	0.355402
$516$	0	0
$517$	0.0198204	0.000871698 0
$518$	0.399212	0.0175404
$519$	0	0
$520$	1.78757	0.0783899
$521$	−1.24188	−0.0544077	−0.0272039	−	0.999630i	$-0.508660\pi$
$521$	−1.24188	−0.0544077	−0.0272039	+	0.999630i	$0.508660\pi$
$522$	0	0
$523$	14.7049	0.643000	0.321500	−	0.946910i	$-0.395813\pi$
$523$	14.7049	0.643000	0.321500	+	0.946910i	$0.395813\pi$
$524$	26.2712	1.14766
$525$	0	0
$526$	2.65257	0.115658
$527$	−17.6900	−0.770587
$528$	0	0
$529$	−21.3710	−0.929175
$530$	−1.48346	−0.0644374
$531$	0	0
$532$	6.84392	0.296722
$533$	38.9755	1.68822
$534$	0	0
$535$	0.366629	0.0158507
$536$	4.49750	0.194262
$537$	0	0
$538$	1.23195	0.0531130
$539$	0.0952401	0.00410228
$540$	0	0
$541$	−30.1754	−1.29734	−0.648670	−	0.761070i	$-0.724674\pi$
$541$	−30.1754	−1.29734	−0.648670	+	0.761070i	$0.724674\pi$
$542$	0.714200	0.0306775
$543$	0	0
$544$	4.76196	0.204167
$545$	23.1627	0.992182
$546$	0	0
$547$	−16.5541	−0.707801	−0.353901	−	0.935283i	$-0.615145\pi$
$547$	−16.5541	−0.707801	−0.353901	+	0.935283i	$0.615145\pi$
$548$	−0.139666	−0.00596622
$549$	0	0
$550$	−0.00634125	−0.000270392 0
$551$	−7.52638	−0.320634
$552$	0	0
$553$	24.7503	1.05249
$554$	1.25362	0.0532613
$555$	0	0
$556$	−11.4084	−0.483822
$557$	−32.8029	−1.38990	−0.694952	−	0.719056i	$-0.744574\pi$
$557$	−32.8029	−1.38990	−0.694952	+	0.719056i	$0.744574\pi$
$558$	0	0
$559$	27.6842	1.17092
$560$	−16.3121	−0.689310
$561$	0	0
$562$	1.26647	0.0534226
$563$	15.6762	0.660671	0.330336	−	0.943864i	$-0.392838\pi$
$563$	15.6762	0.660671	0.330336	+	0.943864i	$0.392838\pi$
$564$	0	0
$565$	6.06999	0.255366
$566$	−0.936717	−0.0393731
$567$	0	0
$568$	−3.39819	−0.142585
$569$	−24.8545	−1.04196	−0.520978	−	0.853570i	$-0.674433\pi$
$569$	−24.8545	−1.04196	−0.520978	+	0.853570i	$0.674433\pi$
$570$	0	0
$571$	−5.04010	−0.210922	−0.105461	−	0.994423i	$-0.533632\pi$
$571$	−5.04010	−0.210922	−0.105461	+	0.994423i	$0.533632\pi$
$572$	−0.163573	−0.00683932
$573$	0	0
$574$	2.90763	0.121362
$575$	4.53919	0.189297
$576$	0	0
$577$	−24.3761	−1.01479	−0.507394	−	0.861714i	$-0.669391\pi$
$577$	−24.3761	−1.01479	−0.507394	+	0.861714i	$0.669391\pi$
$578$	0.240250	0.00999307
$579$	0	0
$580$	18.0121	0.747912
$581$	44.8711	1.86157
$582$	0	0
$583$	0.272042	0.0112668
$584$	3.81137	0.157716
$585$	0	0
$586$	2.77425	0.114603
$587$	−20.4834	−0.845442	−0.422721	−	0.906260i	$-0.638925\pi$
$587$	−20.4834	−0.845442	−0.422721	+	0.906260i	$0.638925\pi$
$588$	0	0
$589$	3.98857	0.164346
$590$	0.919881	0.0378709
$591$	0	0
$592$	5.10375	0.209763
$593$	−34.6141	−1.42143	−0.710716	−	0.703479i	$-0.751629\pi$
$593$	−34.6141	−1.42143	−0.710716	+	0.703479i	$0.751629\pi$
$594$	0	0
$595$	−18.3086	−0.750579
$596$	27.4686	1.12516
$597$	0	0
$598$	−0.475695	−0.0194526
$599$	11.5624	0.472427	0.236213	−	0.971701i	$-0.424094\pi$
$599$	11.5624	0.472427	0.236213	+	0.971701i	$0.424094\pi$
$600$	0	0
$601$	−3.68089	−0.150147	−0.0750733	−	0.997178i	$-0.523919\pi$
$601$	−3.68089	−0.150147	−0.0750733	+	0.997178i	$0.523919\pi$
$602$	2.06528	0.0841746
$603$	0	0
$604$	15.6625	0.637299
$605$	13.2156	0.537290
$606$	0	0
$607$	−19.7099	−0.800000	−0.400000	−	0.916515i	$-0.630990\pi$
$607$	−19.7099	−0.800000	−0.400000	+	0.916515i	$0.630990\pi$
$608$	−1.07368	−0.0435436
$609$	0	0
$610$	0.179138	0.00725309
$611$	4.14315	0.167614
$612$	0	0
$613$	−17.0552	−0.688851	−0.344426	−	0.938814i	$-0.611926\pi$
$613$	−17.0552	−0.688851	−0.344426	+	0.938814i	$0.611926\pi$
$614$	0.447771	0.0180706
$615$	0	0
$616$	−0.0244551	−0.000985324 0
$617$	−11.4384	−0.460492	−0.230246	−	0.973132i	$-0.573953\pi$
$617$	−11.4384	−0.460492	−0.230246	+	0.973132i	$0.573953\pi$
$618$	0	0
$619$	−5.10790	−0.205304	−0.102652	−	0.994717i	$-0.532733\pi$
$619$	−5.10790	−0.205304	−0.102652	+	0.994717i	$0.532733\pi$
$620$	−9.54545	−0.383354
$621$	0	0
$622$	0.175538	0.00703842
$623$	17.5284	0.702261
$624$	0	0
$625$	5.43111	0.217244
$626$	1.17240	0.0468585
$627$	0	0
$628$	−36.8943	−1.47224
$629$	5.72843	0.228408
$630$	0	0
$631$	−14.4135	−0.573794	−0.286897	−	0.957962i	$-0.592624\pi$
$631$	−14.4135	−0.573794	−0.286897	+	0.957962i	$0.592624\pi$
$632$	−2.58682	−0.102898
$633$	0	0
$634$	1.42238	0.0564900
$635$	−8.66228	−0.343752
$636$	0	0
$637$	19.9085	0.788804
$638$	0.0134196	0.000531286 0
$639$	0	0
$640$	3.42370	0.135334
$641$	22.6382	0.894154	0.447077	−	0.894495i	$-0.352465\pi$
$641$	22.6382	0.894154	0.447077	+	0.894495i	$0.352465\pi$
$642$	0	0
$643$	0.258230	0.0101836	0.00509181	−	0.999987i	$-0.498379\pi$
$643$	0.258230	0.0101836	0.00509181	+	0.999987i	$0.498379\pi$
$644$	8.73498	0.344206
$645$	0	0
$646$	−0.398980	−0.0156977
$647$	−3.23831	−0.127311	−0.0636555	−	0.997972i	$-0.520276\pi$
$647$	−3.23831	−0.127311	−0.0636555	+	0.997972i	$0.520276\pi$
$648$	0	0
$649$	−0.168691	−0.00662170
$650$	−1.32554	−0.0519921
$651$	0	0
$652$	−35.1910	−1.37818
$653$	14.7675	0.577898	0.288949	−	0.957345i	$-0.406694\pi$
$653$	14.7675	0.577898	0.288949	+	0.957345i	$0.406694\pi$
$654$	0	0
$655$	15.8460	0.619154
$656$	37.1727	1.45135
$657$	0	0
$658$	0.309085	0.0120494
$659$	12.5721	0.489740	0.244870	−	0.969556i	$-0.421255\pi$
$659$	12.5721	0.489740	0.244870	+	0.969556i	$0.421255\pi$
$660$	0	0
$661$	−15.6029	−0.606883	−0.303442	−	0.952850i	$-0.598136\pi$
$661$	−15.6029	−0.606883	−0.303442	+	0.952850i	$0.598136\pi$
$662$	0.628171	0.0244145
$663$	0	0
$664$	−4.68978	−0.181999
$665$	4.12806	0.160079
$666$	0	0
$667$	−9.60600	−0.371946
$668$	20.6004	0.797051
$669$	0	0
$670$	1.35363	0.0522953
$671$	−0.0328510	−0.00126820
$672$	0	0
$673$	−22.2126	−0.856233	−0.428116	−	0.903724i	$-0.640823\pi$
$673$	−22.2126	−0.856233	−0.428116	+	0.903724i	$0.640823\pi$
$674$	−1.78673	−0.0688223
$675$	0	0
$676$	−8.29765	−0.319140
$677$	−2.67156	−0.102676	−0.0513381	−	0.998681i	$-0.516349\pi$
$677$	−2.67156	−0.102676	−0.0513381	+	0.998681i	$0.516349\pi$
$678$	0	0
$679$	−4.83990	−0.185738
$680$	1.91356	0.0733815
$681$	0	0
$682$	−0.00711165	−0.000272319 0
$683$	−42.5605	−1.62853	−0.814266	−	0.580491i	$-0.802860\pi$
$683$	−42.5605	−1.62853	−0.814266	+	0.580491i	$0.802860\pi$
$684$	0	0
$685$	−0.0842422	−0.00321873
$686$	−0.678390	−0.0259011
$687$	0	0
$688$	26.4037	1.00663
$689$	56.8663	2.16644
$690$	0	0
$691$	8.07955	0.307360	0.153680	−	0.988121i	$-0.450887\pi$
$691$	8.07955	0.307360	0.153680	+	0.988121i	$0.450887\pi$
$692$	44.8682	1.70563
$693$	0	0
$694$	−2.66597	−0.101199
$695$	−6.88119	−0.261018
$696$	0	0
$697$	41.7226	1.58035
$698$	−1.39349	−0.0527442
$699$	0	0
$700$	24.3404	0.919979
$701$	2.74186	0.103558	0.0517792	−	0.998659i	$-0.483511\pi$
$701$	2.74186	0.103558	0.0517792	+	0.998659i	$0.483511\pi$
$702$	0	0
$703$	−1.29159	−0.0487134
$704$	−0.154726	−0.00583147
$705$	0	0
$706$	0.726936	0.0273586
$707$	−33.1647	−1.24729
$708$	0	0
$709$	47.8790	1.79813	0.899066	−	0.437813i	$-0.144247\pi$
$709$	47.8790	1.79813	0.899066	+	0.437813i	$0.144247\pi$
$710$	−1.02277	−0.0383837
$711$	0	0
$712$	−1.83201	−0.0686576
$713$	5.09066	0.190647
$714$	0	0
$715$	−0.0986623	−0.00368976
$716$	12.6053	0.471083
$717$	0	0
$718$	−0.469656	−0.0175274
$719$	8.44756	0.315041	0.157520	−	0.987516i	$-0.449650\pi$
$719$	8.44756	0.315041	0.157520	+	0.987516i	$0.449650\pi$
$720$	0	0
$721$	−23.0648	−0.858979
$722$	0.0899584	0.00334790
$723$	0	0
$724$	1.00917	0.0375057
$725$	−26.7675	−0.994120
$726$	0	0
$727$	−35.5263	−1.31760	−0.658798	−	0.752319i	$-0.728935\pi$
$727$	−35.5263	−1.31760	−0.658798	+	0.752319i	$0.728935\pi$
$728$	−5.11197	−0.189462
$729$	0	0
$730$	1.14712	0.0424570
$731$	29.6355	1.09611
$732$	0	0
$733$	42.6362	1.57480	0.787402	−	0.616439i	$-0.211426\pi$
$733$	42.6362	1.57480	0.787402	+	0.616439i	$0.211426\pi$
$734$	−1.97781	−0.0730023
$735$	0	0
$736$	−1.37035	−0.0505119
$737$	−0.248234	−0.00914380
$738$	0	0
$739$	26.1590	0.962275	0.481138	−	0.876645i	$-0.340224\pi$
$739$	26.1590	0.962275	0.481138	+	0.876645i	$0.340224\pi$
$740$	3.09104	0.113629
$741$	0	0
$742$	4.24231	0.155740
$743$	4.45751	0.163530	0.0817652	−	0.996652i	$-0.473944\pi$
$743$	4.45751	0.163530	0.0817652	+	0.996652i	$0.473944\pi$
$744$	0	0
$745$	16.5682	0.607013
$746$	0.703028	0.0257397
$747$	0	0
$748$	−0.175102	−0.00640235
$749$	−1.04846	−0.0383100
$750$	0	0
$751$	−8.06533	−0.294308	−0.147154	−	0.989114i	$-0.547011\pi$
$751$	−8.06533	−0.294308	−0.147154	+	0.989114i	$0.547011\pi$
$752$	3.95151	0.144097
$753$	0	0
$754$	2.80516	0.102158
$755$	9.44718	0.343818
$756$	0	0
$757$	8.03008	0.291858	0.145929	−	0.989295i	$-0.453383\pi$
$757$	8.03008	0.291858	0.145929	+	0.989295i	$0.453383\pi$
$758$	−0.0218156	−0.000792379 0
$759$	0	0
$760$	−0.431451	−0.0156504
$761$	−24.8207	−0.899751	−0.449876	−	0.893091i	$-0.648532\pi$
$761$	−24.8207	−0.899751	−0.449876	+	0.893091i	$0.648532\pi$
$762$	0	0
$763$	−66.2394	−2.39803
$764$	−40.1424	−1.45230
$765$	0	0
$766$	−2.66850	−0.0964168
$767$	−35.2623	−1.27325
$768$	0	0
$769$	3.13221	0.112950	0.0564751	−	0.998404i	$-0.482014\pi$
$769$	3.13221	0.112950	0.0564751	+	0.998404i	$0.482014\pi$
$770$	−0.00736035	−0.000265249 0
$771$	0	0
$772$	29.4656	1.06049
$773$	35.3644	1.27197	0.635985	−	0.771702i	$-0.280594\pi$
$773$	35.3644	1.27197	0.635985	+	0.771702i	$0.280594\pi$
$774$	0	0
$775$	14.1853	0.509552
$776$	0.505851	0.0181590
$777$	0	0
$778$	−0.685033	−0.0245596
$779$	−9.40722	−0.337049
$780$	0	0
$781$	0.187558	0.00671137
$782$	−0.509223	−0.0182098
$783$	0	0
$784$	18.9877	0.678131
$785$	−22.2536	−0.794264
$786$	0	0
$787$	20.1376	0.717828	0.358914	−	0.933371i	$-0.383147\pi$
$787$	20.1376	0.717828	0.358914	+	0.933371i	$0.383147\pi$
$788$	9.68341	0.344957
$789$	0	0
$790$	−0.778566	−0.0277001
$791$	−17.3586	−0.617200
$792$	0	0
$793$	−6.86700	−0.243854
$794$	−0.0520519	−0.00184725
$795$	0	0
$796$	−55.8871	−1.98087
$797$	−27.8311	−0.985827	−0.492913	−	0.870078i	$-0.664068\pi$
$797$	−27.8311	−0.985827	−0.492913	+	0.870078i	$0.664068\pi$
$798$	0	0
$799$	4.43516	0.156905
$800$	−3.81854	−0.135006
$801$	0	0
$802$	−0.283703	−0.0100179
$803$	−0.210364	−0.00742358
$804$	0	0
$805$	5.26868	0.185697
$806$	−1.48658	−0.0523627
$807$	0	0
$808$	3.46627	0.121943
$809$	17.7922	0.625540	0.312770	−	0.949829i	$-0.398743\pi$
$809$	17.7922	0.625540	0.312770	+	0.949829i	$0.398743\pi$
$810$	0	0
$811$	−46.3954	−1.62916	−0.814581	−	0.580049i	$-0.803033\pi$
$811$	−46.3954	−1.62916	−0.814581	+	0.580049i	$0.803033\pi$
$812$	−51.5099	−1.80764
$813$	0	0
$814$	0.00230292	8.07173e−5 0
$815$	−21.2262	−0.743520
$816$	0	0
$817$	−6.68193	−0.233771
$818$	1.58191	0.0553102
$819$	0	0
$820$	22.5133	0.786200
$821$	42.2504	1.47455	0.737275	−	0.675593i	$-0.236112\pi$
$821$	42.2504	1.47455	0.737275	+	0.675593i	$0.236112\pi$
$822$	0	0
$823$	29.8844	1.04170	0.520852	−	0.853647i	$-0.325614\pi$
$823$	29.8844	1.04170	0.520852	+	0.853647i	$0.325614\pi$
$824$	2.41066	0.0839793
$825$	0	0
$826$	−2.63062	−0.0915310
$827$	−51.8124	−1.80169	−0.900847	−	0.434137i	$-0.857053\pi$
$827$	−51.8124	−1.80169	−0.900847	+	0.434137i	$0.857053\pi$
$828$	0	0
$829$	33.5745	1.16609	0.583045	−	0.812440i	$-0.301861\pi$
$829$	33.5745	1.16609	0.583045	+	0.812440i	$0.301861\pi$
$830$	−1.41150	−0.0489940
$831$	0	0
$832$	−32.3432	−1.12130
$833$	21.3117	0.738406
$834$	0	0
$835$	12.4255	0.430003
$836$	0.0394803	0.00136546
$837$	0	0
$838$	2.85838	0.0987411
$839$	−4.39992	−0.151902	−0.0759511	−	0.997112i	$-0.524199\pi$
$839$	−4.39992	−0.151902	−0.0759511	+	0.997112i	$0.524199\pi$
$840$	0	0
$841$	27.6463	0.953322
$842$	2.83762	0.0977907
$843$	0	0
$844$	−10.3976	−0.357900
$845$	−5.00490	−0.172174
$846$	0	0
$847$	−37.7932	−1.29859
$848$	54.2360	1.86247
$849$	0	0
$850$	−1.41897	−0.0486703
$851$	−1.64848	−0.0565090
$852$	0	0
$853$	−4.56712	−0.156375	−0.0781877	−	0.996939i	$-0.524913\pi$
$853$	−4.56712	−0.156375	−0.0781877	+	0.996939i	$0.524913\pi$
$854$	−0.512288	−0.0175301
$855$	0	0
$856$	0.109582	0.00374543
$857$	−44.6143	−1.52399	−0.761997	−	0.647580i	$-0.775781\pi$
$857$	−44.6143	−1.52399	−0.761997	+	0.647580i	$0.775781\pi$
$858$	0	0
$859$	−7.94120	−0.270950	−0.135475	−	0.990781i	$-0.543256\pi$
$859$	−7.94120	−0.270950	−0.135475	+	0.990781i	$0.543256\pi$
$860$	15.9912	0.545295
$861$	0	0
$862$	1.11830	0.0380894
$863$	32.7772	1.11575	0.557874	−	0.829926i	$-0.311617\pi$
$863$	32.7772	1.11575	0.557874	+	0.829926i	$0.311617\pi$
$864$	0	0
$865$	27.0632	0.920177
$866$	1.61989	0.0550463
$867$	0	0
$868$	27.2975	0.926537
$869$	0.142776	0.00484335
$870$	0	0
$871$	−51.8895	−1.75821
$872$	6.92313	0.234447
$873$	0	0
$874$	0.114815	0.00388367
$875$	35.3217	1.19409
$876$	0	0
$877$	−51.2962	−1.73215	−0.866074	−	0.499915i	$-0.833365\pi$
$877$	−51.2962	−1.73215	−0.866074	+	0.499915i	$0.833365\pi$
$878$	0.210518	0.00710464
$879$	0	0
$880$	−0.0940988	−0.00317207
$881$	−33.5440	−1.13013	−0.565064	−	0.825047i	$-0.691148\pi$
$881$	−33.5440	−1.13013	−0.565064	+	0.825047i	$0.691148\pi$
$882$	0	0
$883$	−21.9754	−0.739532	−0.369766	−	0.929125i	$-0.620562\pi$
$883$	−21.9754	−0.739532	−0.369766	+	0.929125i	$0.620562\pi$
$884$	−36.6024	−1.23107
$885$	0	0
$886$	0.840557	0.0282391
$887$	−26.8345	−0.901013	−0.450506	−	0.892773i	$-0.648757\pi$
$887$	−26.8345	−0.901013	−0.450506	+	0.892773i	$0.648757\pi$
$888$	0	0
$889$	24.7719	0.830821
$890$	−0.551388	−0.0184826
$891$	0	0
$892$	−52.8259	−1.76874
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	7.60316	0.254146
$896$	−9.79090	−0.327091
$897$	0	0
$898$	−1.20667	−0.0402672
$899$	−30.0195	−1.00121
$900$	0	0
$901$	60.8744	2.02802
$902$	0.0167731	0.000558484 0
$903$	0	0
$904$	1.81426	0.0603415
$905$	0.608705	0.0202340
$906$	0	0
$907$	37.4937	1.24496	0.622479	−	0.782637i	$-0.286126\pi$
$907$	37.4937	1.24496	0.622479	+	0.782637i	$0.286126\pi$
$908$	−3.15620	−0.104742
$909$	0	0
$910$	−1.53857	−0.0510031
$911$	45.6228	1.51155	0.755776	−	0.654831i	$-0.227260\pi$
$911$	45.6228	1.51155	0.755776	+	0.654831i	$0.227260\pi$
$912$	0	0
$913$	0.258846	0.00856656
$914$	1.71102	0.0565955
$915$	0	0
$916$	−8.91052	−0.294412
$917$	−45.3154	−1.49645
$918$	0	0
$919$	−1.57412	−0.0519255	−0.0259627	−	0.999663i	$-0.508265\pi$
$919$	−1.57412	−0.0519255	−0.0259627	+	0.999663i	$0.508265\pi$
$920$	−0.550666	−0.0181549
$921$	0	0
$922$	−0.231399	−0.00762072
$923$	39.2063	1.29049
$924$	0	0
$925$	−4.59355	−0.151035
$926$	2.86694	0.0942134
$927$	0	0
$928$	8.08094	0.265270
$929$	23.1389	0.759163	0.379582	−	0.925158i	$-0.376068\pi$
$929$	23.1389	0.759163	0.379582	+	0.925158i	$0.376068\pi$
$930$	0	0
$931$	−4.80516	−0.157483
$932$	−12.5242	−0.410244
$933$	0	0
$934$	2.00852	0.0657207
$935$	−0.105616	−0.00345402
$936$	0	0
$937$	25.7216	0.840287	0.420143	−	0.907458i	$-0.361980\pi$
$937$	25.7216	0.840287	0.420143	+	0.907458i	$0.361980\pi$
$938$	−3.87103	−0.126394
$939$	0	0
$940$	2.39320	0.0780575
$941$	−4.56547	−0.148830	−0.0744150	−	0.997227i	$-0.523709\pi$
$941$	−4.56547	−0.148830	−0.0744150	+	0.997227i	$0.523709\pi$
$942$	0	0
$943$	−12.0065	−0.390987
$944$	−33.6313	−1.09461
$945$	0	0
$946$	0.0119139	0.000387355 0
$947$	−35.5165	−1.15413	−0.577065	−	0.816698i	$-0.695802\pi$
$947$	−35.5165	−1.15413	−0.577065	+	0.816698i	$0.695802\pi$
$948$	0	0
$949$	−43.9734	−1.42744
$950$	0.319936	0.0103801
$951$	0	0
$952$	−5.47227	−0.177357
$953$	−36.4091	−1.17941	−0.589704	−	0.807620i	$-0.700755\pi$
$953$	−36.4091	−1.17941	−0.589704	+	0.807620i	$0.700755\pi$
$954$	0	0
$955$	−24.2127	−0.783506
$956$	−15.7867	−0.510577
$957$	0	0
$958$	2.61836	0.0845953
$959$	0.240911	0.00777941
$960$	0	0
$961$	−15.0913	−0.486816
$962$	0.481391	0.0155207
$963$	0	0
$964$	33.2054	1.06948
$965$	17.7728	0.572127
$966$	0	0
$967$	−18.0804	−0.581428	−0.290714	−	0.956810i	$-0.593893\pi$
$967$	−18.0804	−0.581428	−0.290714	+	0.956810i	$0.593893\pi$
$968$	3.95002	0.126958
$969$	0	0
$970$	0.152248	0.00488839
$971$	−28.5303	−0.915581	−0.457790	−	0.889060i	$-0.651359\pi$
$971$	−28.5303	−0.915581	−0.457790	+	0.889060i	$0.651359\pi$
$972$	0	0
$973$	19.6784	0.630861
$974$	−0.0913381	−0.00292666
$975$	0	0
$976$	−6.54938	−0.209640
$977$	54.9761	1.75884	0.879421	−	0.476044i	$-0.157930\pi$
$977$	54.9761	1.75884	0.879421	+	0.476044i	$0.157930\pi$
$978$	0	0
$979$	0.101116	0.00323167
$980$	11.4997	0.367345
$981$	0	0
$982$	1.28951	0.0411499
$983$	−24.8556	−0.792772	−0.396386	−	0.918084i	$-0.629736\pi$
$983$	−24.8556	−0.792772	−0.396386	+	0.918084i	$0.629736\pi$
$984$	0	0
$985$	5.84075	0.186102
$986$	3.00287	0.0956310
$987$	0	0
$988$	8.25277	0.262556
$989$	−8.52822	−0.271182
$990$	0	0
$991$	24.1322	0.766583	0.383292	−	0.923627i	$-0.374790\pi$
$991$	24.1322	0.766583	0.383292	+	0.923627i	$0.374790\pi$
$992$	−4.28246	−0.135968
$993$	0	0
$994$	2.92484	0.0927704
$995$	−33.7095	−1.06866
$996$	0	0
$997$	−2.87119	−0.0909316	−0.0454658	−	0.998966i	$-0.514477\pi$
$997$	−2.87119	−0.0909316	−0.0454658	+	0.998966i	$0.514477\pi$
$998$	−1.30193	−0.0412118
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.18	yes	34
3.2	odd	2		8037.2.a.v.1.17	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.17	✓	34	3.2	odd	2
8037.2.a.w.1.18	yes	34	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0899584 q^{2} -1.99191 q^{4} -1.20146 q^{5} +3.43586 q^{7} -0.359105 q^{8} +O(q^{10})\)	\(q+0.0899584 q^{2} -1.99191 q^{4} -1.20146 q^{5} +3.43586 q^{7} -0.359105 q^{8} -0.108081 q^{10} +0.0198204 q^{11} +4.14315 q^{13} +0.309085 q^{14} +3.95151 q^{16} +4.43516 q^{17} -1.00000 q^{19} +2.39320 q^{20} +0.00178301 q^{22} -1.27631 q^{23} -3.55649 q^{25} +0.372711 q^{26} -6.84392 q^{28} +7.52638 q^{29} -3.98857 q^{31} +1.07368 q^{32} +0.398980 q^{34} -4.12806 q^{35} +1.29159 q^{37} -0.0899584 q^{38} +0.431451 q^{40} +9.40722 q^{41} +6.68193 q^{43} -0.0394803 q^{44} -0.114815 q^{46} +1.00000 q^{47} +4.80516 q^{49} -0.319936 q^{50} -8.25277 q^{52} +13.7254 q^{53} -0.0238134 q^{55} -1.23384 q^{56} +0.677060 q^{58} -8.51100 q^{59} -1.65744 q^{61} -0.358806 q^{62} -7.80643 q^{64} -4.97783 q^{65} -12.5242 q^{67} -8.83444 q^{68} -0.371353 q^{70} +9.46292 q^{71} -10.6135 q^{73} +0.116190 q^{74} +1.99191 q^{76} +0.0681001 q^{77} +7.20351 q^{79} -4.74758 q^{80} +0.846258 q^{82} +13.0596 q^{83} -5.32867 q^{85} +0.601095 q^{86} -0.00711760 q^{88} +5.10160 q^{89} +14.2353 q^{91} +2.54229 q^{92} +0.0899584 q^{94} +1.20146 q^{95} -1.40864 q^{97} +0.432265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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